Mirrored geometry does not receive the same material?
Suggestion

Apply the mirror, separate the eyes, transform origin to geometry, uv map, and apply a circular texture.

I do not think attempting to apply the same texture space of [Object] to both eyes in an easy or flexible idea.  If the eyes rotate you may find trouble.  I do not know your animation style.



Secondary Information , Not a suggestion

Here is deprecated texture mapping without uv with typical sticky style ... it is biased to the vector indicated 



If you choose to have bad frozen eyes with a slippery texture this is a bad deprecated node image.  



Both images above rely on the spherical shape and have no uv map required.